Online Bidding Algorithms with Strict Return on Spend (ROS) Constraint

TL;DR

This paper studies online ad bidding algorithms under a strict return-on-spend constraint, revealing fundamental limitations and proposing near-optimal strategies despite the problem's inherent complexity.

Contribution

It introduces the ROSC problem, proves an impossibility result for sub-linear regret, and develops algorithms with near-optimal regret guarantees.

Findings

No online algorithm can achieve sub-linear regret under ROSC.

The problem remains challenging even with constant values.

An algorithm with regret close to the theoretical lower bound is proposed.

Abstract

Auto-bidding problem under a strict return-on-spend constraint (ROSC) is considered, where an algorithm has to make decisions about how much to bid for an ad slot depending on the revealed value, and the hidden allocation and payment function that describes the probability of winning the ad-slot depending on its bid. The objective of an algorithm is to maximize the expected utility (product of ad value and probability of winning the ad slot) summed across all time slots subject to the total expected payment being less than the total expected utility, called the ROSC. A (surprising) impossibility result is derived that shows that no online algorithm can achieve a sub-linear regret even when the value, allocation and payment function are drawn i.i.d. from an unknown distribution. The problem is non-trivial even when the revealed value remains constant across time slots, and an algorithm with regret guarantee that is optimal up to logarithmic factor is derived.